Theme

The comma that will not close

Twelve fifths do not equal seven octaves. Every tuning system in history is a different decision about where to hide the difference.

Twelve fifths do not make seven octaves. Pitch drawn as a spiral: one turn is one octave, so a note's angle is its pitch class and its radius is how high it has climbed. Twelve pure fifths wind round 7 times and a little further, finishing 23.46 cents past seven octaves — the Pythagorean comma. Nothing in the chain closes that gap; it is what a tuning has to absorb, hide or spread.

Pitch and tuning

Twelve fifths and seven octaves, which are not the same thing

Stack twelve perfect fifths and the note that arrives should be the one seven octaves up. It is sharp by about a quarter of a semitone, and the whole history of tuning is a set of decisions about what to do with that.

How far each system sits from equal temperament. Deviation from equal temperament, in cents, for each degree of the chromatic scale. Zero is equal temperament by definition; the just major third is about fourteen cents flat of it and the Pythagorean major third about eight cents sharp.

Pitch and tuning

Where to hide the comma, which is the only real question

Four tuning systems, four different places to put an error that cannot be removed. Each one is coherent, each was standard somewhere, and the choice between them decided what music could be written.

Pitch and tuning

The wolf at the end of the chain

Put all the tuning error into one interval and eleven others become perfect. The twelfth becomes a howl, and for two centuries keyboards were built that way on purpose.

The syntonic comma on the lattice. Pitch classes reached by multiplying by three — a fifth, one step east — and by five — a major third, one step north. Two routes to the same note name arrive at different pitches, and the syntonic comma is the size of that disagreement: 21.51 cents.

Pitch and tuning

A second comma, arriving by a different road

Four pure fifths ought to land on a pure major third, two octaves up. They miss by 21.5 cents — a different gap from the one twelve fifths leave, produced by a different route, and the two are not the same size.

Key character in Werckmeister III. The twelve major keys in circle-of-fifths order, each with a bar as long as its major third is sharp of a pure 5:4. The bars run from 3.9 to 21.5 cents, so the keys genuinely differ.

Pitch and tuning

Keys that had characters, and could be measured

Eighteenth-century writers described E flat major as devotional and F sharp major as harsh, and modern readers file it under synaesthesia. On the instruments those writers used, the difference between the two keys was seventeen cents of third, and that is a quantity.

Every equal division from 5 to 60, and how wrong it is. For each number of equal steps in the octave, how far its best fifth and its best major third fall from the pure ratios, in cents. The divisions people have actually used are the ones with small errors in both, and no other criterion was applied to pick them out.

Pitch and tuning

Nineteen, thirty-one and fifty-three

Divide the octave into a different number of equal parts and different problems disappear. Which ones vanish is not a matter of taste — it can be computed from the prime factors of the intervals, and each division makes a different decision.

220 Hz against 223 Hz. Two tones 3 hertz apart, added. The rapid oscillation is their average; the slow swelling is their difference, heard as 3 beats a second and used by every tuner who has ever worked by ear.

Intervals and chords

Beats are arithmetic that anybody can hear

Two tones a few hertz apart swell and fade at exactly their difference. It is the most direct evidence available that the ear does sums on what reaches it, and it is how every instrument in the world gets tuned.

The circle of fifths. The twelve pitch classes arranged so that each is a fifth above the last. Keys next to each other differ by one sharp or flat, which is why the notes of a key form a contiguous arc rather than a scattering.

Scales and modes

Keys are neighbours, and the map is computed

Two keys a fifth apart share six of their seven notes. That single fact generates the circle of fifths, the key signatures, the shape of nearly every modulation, and the sense that some keys are far away.

The seven chords of a key, by distance from home. Each triad of the major scale placed at a radius equal to how far its voices must move from the tonic chord. The chords that feel closest to home are the ones that are closest, in the plain arithmetic sense.

Harmony and voice leading

A progression is a path, and the map can be drawn

Chords in a key are not a list. They sit at measurable distances from home, and a progression is a walk across that space — which explains why some sequences feel like journeys and others like wandering.

Where a progression in pure ratios ends up. A short chord sequence taken in exact whole-number ratios, with the running difference from equal temperament measured in cents. The sequence returns to its starting chord and the pitch does not, which is the comma arriving in ordinary music.

Harmony and voice leading

The progression that never comes home

Just intonation is usually said to fail because a keyboard has only twelve keys. That is not the reason. Take one of the commonest chord sequences in tonal music, play every chord in exact whole-number ratios, and it arrives back on its starting chord a syntonic comma flat — on any instrument, including a voice.

The smallest audible difference, and what has to clear it. The difference limen for frequency, converted from Wier, Jesteadt and Green's 1977 fit into cents, against the intervals and commas the rest of these essays argue about. Anything drawn below the curve is a quantity nobody can hear as a change of pitch; anything well above it is a quantity a listener can be asked about. The limen is for pure tones, successive, with trained listeners — the most favourable case there is, and therefore the right one to test a claim against.

Perception and the listener

How small a difference is audible

Every essay here about tuning has assumed a listener who can hear the difference between two systems. The assumption has a number: about five cents in the middle of the range. It clears the two commas fourfold and it does not clear the schisma at all, which sorts the whole subject of tuning into the part that is about music and the part that is about arithmetic.

Perception and the listener

The chord is still major, and that is why temperament works

A major third can be seventeen cents wrong and still be a major third. That tolerance is not a failure of hearing — it is the reason the whole subject of tuning is a discussion rather than a catastrophe. Every temperament ever proposed moves intervals around inside their categories, and the one thing none of them may do is push one across a boundary.

Pitch and tuning

A guitar cannot be in tune

Three errors land on the same instrument — equal temperament's own thirds, the sharpening that comes from pressing a string down to a fret, and the fact that the only correction available is one length per string. Searching over every setting a luthier could choose leaves a worst-case error of about fifteen cents, and most of what is left is the temperament, which no saddle can reach.

Every keyboard temperament before 1800, on one line. The fifth's departure from pure against the major third's, for a temperament that narrows every fifth by the same amount. The relation is a straight line — a third is four fifths — so the systems that are usually presented as rival schemes are points on one continuum, and the parameter that indexes them is a single number.

Pitch and tuning

A fraction of a comma

The tuning systems of three centuries are usually presented as rival schemes with names and dates. They are points on a line. Narrow every fifth by the same fraction of a comma and both errors that matter are linear in that fraction — and the system named as the break with the tradition turns out to be a member of it, at one part in eleven.

The same total, distributed differently. The departure of each of the twelve major thirds from a pure 5:4, one row per temperament, keys ordered round the circle of fifths. The totals are equal — every closing temperament's twelve thirds add to 4800 cents — so what a temperament chooses is not how much error there is but which keys carry it.

Pitch and tuning

What a temperament cannot do

Add up the twelve major thirds of any keyboard tuning whose chain of fifths closes and the answer is 4,800 cents. Every time, in every system, before a single fifth has been chosen. So no temperament has ever made the average third less wrong than any other one, and four hundred years of argument were about a distribution rather than about accuracy.

The comma is conserved; only the place it is paid is chosen. Every policy available to a continuous-pitch ensemble on one line: drift per circuit along the bottom, worst mistuned interval up the side. The line is straight and its intercepts are fixed, because the 4 moves of the pump have to absorb 21.51 cents between them however they are shared. Spreading it evenly puts 5.377 cents on each interval — the same narrowing quarter-comma meantone applies to every fifth.

Harmony and voice leading

Somebody has to pay the comma

A choir has no frets and no keys, so it can tune every chord exactly — and a common progression sung that way arrives a comma flat, every circuit. The freedom a keyboard lacks turns out to be a freedom about where the error goes rather than whether there is one, and the two costs always add to the same number.

Nine commas, and the line under which none of them matters. Each named comma at its true size in cents, against the difference limen of 3.87 cents at 500 Hz — the smallest change of frequency a listener can detect, computed from the same formula the perception essays use. One comma falls below it, and it is the only gap in the subject that no tuning system has to do anything about.

Pitch and tuning

A comma under the threshold

Eight pure fifths taken downwards arrive at a major third 1.95 cents flat of a perfect 5:4. That gap has a name and a history, and it is the only one in the subject nobody has ever had to hide — it is under the smallest difference a listener can detect, which makes a chain of untempered fifths a source of almost-just thirds and makes one eighteenth-century tuning nearly free.

Scales and modes

Where the chain was never closed

Eleven earlier essays treat the comma as a fact about music. It is a fact about three requirements — that the scale is built from fifths, that the octave stays pure, and that the chain closes so any degree can be a tonic — and most of the world's tuning traditions decline at least one of them. What they pay instead is computable, and it is paid somewhere else entirely.

How far a chain of fifths is from closing, after each number of fifths. The distance from a whole number of octaves after each number of pure fifths, in cents, out to 60. It is never zero, because a fifth is 3/2 and no power of 3 is a power of 2. The running minima are 1, 2, 5, 12, 41, 53 fifths, and the residue at each is that division's comma: 23.46 cents at 12, 19.84 cents at 41, 3.62 cents at 53. A 3-against-2 polyrhythm asked the same question closes exactly and at once — 3 pulses of the 3 are 2 of the 2, to the last microsecond — because 3/2 is a ratio of whole numbers and log2(3/2) is not.

Pitch and tuning

A comma is a polyrhythm that never closes

Seven earlier essays have rested on a common grid. Two cycles have one when their ratio is a ratio of whole numbers, and the fifth against the octave is not — provably, from unique factorisation. So there is no grid, no composite and no least common multiple, and what is left over instead is the sequence 2, 5, 12, 41, 53 with a comma attached to each.

The sizes a chain of a pure fifth will make. Chains of a pure fifth of two to 12 notes, each folded into one octave and drawn on a 1200-cent line. A chain has exactly two step sizes at 2, 3, 5, 7, 12 notes and three or more at every other size in this range. The five-note and seven-note cases are the chain's pentatonic and its diatonic set, and they are neighbours in a series rather than two separate facts.

Scales and modes

The only sizes a fifth will make

Stack pure fifths and fold them into an octave, and at almost every number of notes the result has three or more different step sizes. At 2, 3, 5, 7, 12, 17 and 29 it has exactly two — and at no other size below thirty. The pentatonic and the diatonic set are not two discoveries. They are neighbours in one series, and it is the comma's series.

A note sung at 440 Hz, drawn in cents. Deviation from the notated pitch against time, for a vibrato of ±71 cents at 6.0 cycles a second — Seashore's and Prame's measured values, which agree. The note is 142 cents wide, and the shaded bands across the middle are the syntonic comma at 21.5 cents, the Pythagorean comma at 23.5 cents, the difference limen at 440 Hz at 4.0 cents. The pitch a listener reports is near the middle of the excursion rather than at either edge — and not exactly at the middle either: because cents are logarithmic and frequency is not, the mean frequency of this trace sits 0.73 cents above the centre line.

Pitch and tuning

A note that is never at its pitch

Eleven essays in this field argue about differences of one to twenty-four cents. An ordinary operatic vibrato is a hundred and forty cents wide and completes six excursions a second, so every one of those distinctions fits inside a single sung note several times over — and the beat rate a tuner would null passes through zero eleven times a second.

Pitch and tuning

A wind instrument is a thermometer

Pitch goes as the square root of absolute temperature, so a warming clarinet sharpens by about three cents a degree and passes a Pythagorean comma after eight. The strings beside it go flat as they warm. Nobody chose either number, no temperament addresses either of them, and together they are twice the size of the discrepancy this whole field is named after.

Pitch and tuning

The note that is sharp because of where it goes

Eleven essays here have tuned notes against other notes sounding at the same time. A melodic line makes the opposite demand of the same note. In C major the leading note wants to be 1088 cents if the argument is vertical — a pure third above the dominant — and 1110 if it is horizontal, because a leading note is a note that is about to arrive somewhere and performers narrow the step. The two answers differ by 21.5 cents, which is a syntonic comma exactly, and no keyboard has ever been able to hold both.

Intervals and chords

The same distance, under two names

Four hundred cents is a major third or a diminished fourth, and on a keyboard nothing in the sound distinguishes them. An earlier essay was about the boundary between two categories; this is about two categories at one acoustic value, and the surprise is where the ambiguity comes from. In quarter-comma meantone a major third is 386 cents and a diminished fourth is 427 — two names, two pitches, forty-one cents apart. Equal temperament collapsed them, and what a listener now supplies from context used to be in the sound.

$Where the two spellings meet, and where they cross. G♯ minus A♭ against the fraction of a comma each fifth is narrowed by — twelve fifths against seven octaves, and nothing else in the calculation. In Pythagorean tuning G♯ is 23.46 cents ABOVE A♭; in quarter-comma meantone it is 41.06 cents BELOW it; the two spellings coincide at 0.09090 of a comma, which is what equal temperament is. A page that distinguishes the two names is exact in every tuning on this line except one point on it, and at that point it is wrong by 0.0014 cents rather than by nothing, because one eleventh is not quite the crossing.$ Pitch and tuning

Two names for one key

A keyboard has one key between G and A and the page has two names for it. That looks like redundancy and it is not: the two names are twelve fifths apart on a chain, and in every tuning anybody played before the nineteenth century they are two different pitches. The size of the difference is twelve fifths against seven octaves and nothing else — twenty-three cents one way in Pythagorean, forty-one the other way in meantone, and zero at exactly one point in between.

How far equal temperament puts each interval's coincidence out. For each interval, the pair of partials it brings together and how many cents equal temperament mistunes that coincidence by. A fifth's third-against-second is out by 2 cents and a major third's fifth-against-fourth by 14 — so the same temperament that is inaudible on a fifth produces, at 220 hertz, a beat of 8.7 per second between two sections singing a third, with every singer in both of them perfectly in tune.

Timbre and acoustics

A section against another section

The choir has been treated as a unison, and no choir sings only unisons. Two sections an interval apart beat between partials rather than between fundamentals — the third brings the fifth partial of one against the fourth of the other — and equal temperament puts that coincidence fourteen cents out. So two sections singing a tempered third beat at nearly nine per second with every singer in both of them perfectly in tune, and the same temperament is inaudible on a fifth.

The beat rate between two sections, second by second. The 5th partial of the lower section against the 4th of the upper, over 256 pairs of voices each sweeping 100 cents 6 times a second with its own phase. The band is the tenth to the ninetieth percentile of the instantaneous rate and the line is the median. With no vibrato the whole thing would be one flat line at 8.7 hertz, which is what was computed earlier. With it, the pair is inside the beating band 20 per cent of the time and above it for the rest — so what a listener gets is neither a beat nor a roughness but an alternation between them at the vibrato rate.

Intervals and chords

Sixteen sweeps against sixteen

Every intonation figure about the voice treats a singer as a frequency. A singer is a frequency being swept a hundred cents wide six times a second, and two sections singing an interval are two hundred and fifty-six pairs of sweeps. The beat rate between the partials the interval brings together stops being a number and becomes a function of time — and the pair spends four fifths of its time above the rate at which beating is beating at all.

The ensemble agrees with itself more and more, about a pitch that is moving. Runs of 16 players each correcting toward the mean of their neighbours, with no term anywhere pulling them back to the note they were given, over 480 corrections. The shaded band is the root-mean-square displacement across all eight runs — the envelope a random walk has — and it grows from 0.84 cents a quarter of the way through to 2.01 at the end, which is the square-root growth a random walk has. Four individual runs are drawn inside it and the furthest of the eight over the top, ending at 4.16 cents. Meanwhile the spread AMONG the players falls from 2.8 cents to 0.5. A consensus with no anchor cannot hold a pitch, and it also cannot lose one quickly: a movement's worth of corrections is a few cents rather than the semitone unaccompanied choirs are said to fall by.

Pitch and tuning

A consensus with nothing to hold it

Once the oboe has stopped, no reference is left in the room. Each player corrects toward what they hear around them, which is other players correcting toward them — and a consensus dynamic has a fixed point at every common value, so it pulls the ensemble together and nothing pulls it anywhere in particular. Simulated, the players' spread falls from 2.8 cents to 0.5 while the ensemble as a whole random-walks. The size is the result and it is small: two or three cents over a movement, which is a tenth of what unaccompanied choirs are said to lose.

A string quartet's open strings are five keys of a Pythagorean keyboard. The five pitch classes a string quartet's open strings sound — C, G, D, A and E — laid out as the chain of fifths they are tuned along, outward from the A the ensemble is given, with each fifth pure. The bars give each string's departure from the same note on an equal-tempered keyboard: C −5.87 cents, G −3.91 cents, D −1.96 cents, A 0.00 cents, E +1.96 cents. Above, the strings each instrument owns: the violin G, D, A, E; the viola C, G, D, A; the cello the same four an octave lower. The cello's C2 is 0.221 hertz below the keyboard's, and the widest span of the chain, from the cello's C to the violin's E, is a Pythagorean third and two octaves, 21.5 cents wider than a just one.

Pitch and tuning

The tuning a string quartet cannot change

A string quartet can put every stopped note wherever it likes, and it has five pitches it cannot move once the pegs are turned: the open strings C, G, D, A and E, tuned in pure fifths from the A it was given. That makes its only fixed tuning a five-key Pythagorean keyboard — the cello's C nearly six cents below a piano's, and every third two open strings can make a syntonic comma from just. Counted key by key, the clash is worst in G, C and F, the keys that use every open string, and absent from A and E.

A tempered fifth is a beat a second on the violin and one in four and a half seconds on the cello. How fast each open fifth of a string quartet beats when it is narrowed by 1.955 cents, the narrowing that meets an equal-tempered keyboard. The beat is the lower string's third partial against the upper string's second, so it is proportional to the lower string's frequency. Cello C2–G2: 0.22 a second, one beat every 4.5 seconds; cello G2–D3: 0.33 a second, one beat every 3.0 seconds; cello D3–A3: 0.50 a second, one beat every 2.0 seconds; viola C3–G3: 0.44 a second, one beat every 2.3 seconds; viola G3–D4: 0.66 a second, one beat every 1.5 seconds; violin G3–D4: 0.66 a second, one beat every 1.5 seconds; violin D4–A4: 1.00 a second, one beat every 1.0 seconds; violin A4–E5: 1.49 a second, one beat every 0.7 seconds. The slowest, the cello's C2–G2, is 6.7 times slower than the violin's A4–E5.

Pitch and tuning

The cello cannot hear its own tempering

Narrowing a quartet's fifths to meet a piano is one number, 1.96 cents a fifth, and it is a different beat on every string: once every two thirds of a second on the violin's A–E and once every four and a half seconds on the cello's C–G. Set by ear for two seconds a fifth, the violin's E lands within two thirds of a cent and the cello's C within 5.7 — which is as large as the Pythagorean error the tempering was meant to remove. The string whose tuning is most wrong is the string whose tuning is least certain, and a cellist tuning down the chain cannot tell pure from tempered.

Pitch and tuning

An open string pulls the quartet flat

Once the tuning note has stopped, a quartet corrects toward itself and nothing holds its pitch. But four of its pitches do not move: the open strings, on a Pythagorean chain from C 5.9 cents flat to E 2.0 sharp, each ringing when a stopped note shares its pitch class. Give that sympathy a weight of a hundredth of a correction and it beats the random walk within a movement. The quartet settles flat in every major key — by 0.9 cents in E, 2.5 in C and 2.6 in A♭ — and in A♭ major the cellist's tuning scatter moves the whole ensemble by 1.7 cents.

Four pure fourths and a pure third leave a comma on the top strings. Each of a guitar's six open strings, in cents from the note its own frets make, when the strings are tuned against each other three ways. fretted unisons: E2 +0.00, A2 +0.00, D3 +0.00, G3 +0.00, B3 +0.00, E4 +0.00. harmonics, pure third on G–B: E2 +0.00, A2 −1.96, D3 −3.91, G3 −5.87, B3 −19.55, E4 −21.51. harmonics, B from the low E's twelfth: E2 +0.00, A2 −1.96, D3 −3.91, G3 −5.87, B3 +1.96, E4 +0.00. Tuned with pure fourths and a pure third the high E closes two octaves 21.51 cents flat, a syntonic comma; taking the B from the low E's twelfth lands the high E exactly and moves the error into the third between the G and B strings.

Pitch and tuning

A guitar tuned by harmonics hides a comma

A quartet's stopped notes go wherever its players put them, so its open strings can sit on a pure chain. A guitar's stopped notes are fixed by frets that are already an equal temperament, and its six open strings — four fourths and a third — close two octaves exactly a syntonic comma short when tuned pure, since (4/3)⁴·(5/4) is 4 × 80/81. Tuning by harmonics decides where the comma goes, not whether: a pure third puts it in the octaves of an open E major chord, 17.6 cents narrow; a B from the low E's twelfth puts it in the G–B third, 21.5 cents wide.

One of these tunings has no comma to place. Five guitar tunings, each with every adjacent interval set to the pure ratio nearest the interval it spans, drawn as how far each string then sits from where the frets put it. standard, spanning 5–5–5–4–5 semitones, closes 21.5 cents short and spreads its strings over 21.5 cents; drop D, spanning 7–5–5–4–5 semitones, closes 17.6 cents short and spreads its strings over 19.6 cents; D A D G A D, spanning 7–5–5–2–5 semitones, closes exactly and spreads its strings over 3.9 cents; open G, spanning 5–7–5–4–3 semitones, closes exactly and spreads its strings over 15.6 cents; open D, spanning 7–5–4–3–5 semitones, closes exactly and spreads its strings over 15.6 cents. Standard tuning's four fourths and a third fall a syntonic comma short of two octaves, which is what was found earlier. D A D G A D's fifth, two fourths, a tone and a fourth multiply to exactly four — two octaves — so nothing has to be put anywhere.

Pitch and tuning

One tuning has no comma to place

Standard guitar tuning is four fourths and a third, and tuned pure it closes two octaves exactly a syntonic comma short — so tuning by ear decides where the comma goes rather than whether. An open tuning replaces the interval list. D A D G A D's fifth, two fourths, a tone and a fourth multiply to exactly four, so its chain closes and there is nothing to place: its six strings sit within four cents of the frets. The open-chord tunings close too, and pay for it with one string fifteen cents flat — which every chord barred straight across inherits exactly, and which is why every such chord is precisely as pure as the open one.

Pitch and tuning

A tuning is right for some chords and wrong for the rest

An earlier essay asked which tuning minimises a piece's total departure from just, and guessed the answer would be a few cents off a named one. Searched over all six strings, two of three pieces get back the tuning they were already in — because their chains close and there is nothing to improve. The third lands on a tuning nobody names, three and nine tenths of a cent better than anything a player would have tried, and it is not a compromise: two of its three chords are exactly just and the third is abandoned by thirteen cents.

Counted in beats, the abandoned chord comes back sooner and by steps. The piece in standard tuning with its G major, open chord given more and more of the piece, searched twice: once minimising the mean departure from just in cents, once minimising the mean beat rate. The vertical axis is how far that chord sits from just at each optimum. Counted in cents it stays at 13.29 cents until it holds exactly 44 beats — the same as the rest of the piece — and then drops to zero in one step. Counted in beats it drops at 30.4 beats to 4.96, then at 47.4 beats to 3.91, then at 73.1 beats to 0.00 cents. At the piece's own 8 beats the two counts agree exactly: the corner does not move, and what the beats change is where the steps are.

Pitch and tuning

Counting beats moves the price of a chord, not the tuning

The search that found a guitar piece's best tuning counted every cent of error alike, and the obvious objection is that a cent of a third beats faster than a cent of an octave. Counted in beats instead, every piece gets exactly the same tuning back. What changes is how much of the piece the abandoned chord has to hold before it is rescued — thirty beats instead of forty-four, arriving in three steps instead of one.

Two shapes asking fifteen pairs of strings for different things. Every pair of the six strings, and the difference in cents between the two strings' offsets that would make the interval each shape puts on that pair exactly just. The piece's chords fall into two families that never disagree with themselves: E major, open and A major barred at 5 (44 beats), and G major, open (8 beats). They disagree on 13 of the 15 pairs: E2–A2 is asked for 1.96 and -13.69; E2–D3 is asked for 0.00 and 1.96; E2–G3 is asked for -13.69 and 0.00; E2–B3 is asked for 1.96 and -13.69; A2–D3 is asked for -1.96 and 15.64; A2–G3 is asked for -15.64 and 13.69; A2–E4 is asked for -1.96 and 13.69; D3–G3 is asked for -13.69 and -1.96; D3–B3 is asked for 1.96 and -15.64; D3–E4 is asked for 0.00 and -1.96; G3–B3 is asked for 15.64 and -13.69; G3–E4 is asked for 13.69 and 0.00; B3–E4 is asked for -1.96 and 13.69. The ring on each row is where the piece's cheapest tuning actually puts the pair. It sits on the first family's demand every time, which is what abandoning the other chord means: no weighting of the error can put a ring on two different places.

Pitch and tuning

The chord a tuning gives up is a fingering

A guitar piece's best tuning makes two of its chords exactly just and abandons the third by thirteen cents, and no way of counting error — cents, beats, squared cents, or only the beats slow enough to hear — brings the third chord back. Fingered as a barre chord in the shape the other two already use, it comes back completely, and the whole piece is exactly just. The conflict was never between chords. It was between hands.

A tempered major third and its own ghost are not one sound. The multiple of the implied fundamental that each reading names, against how far the upper note is tempered from the just major third, at 55 and 80 dB. At 55 dB the notes and products together name the fundamental from −9.0 cents to +8.5 cents and no template at all outside that; the products alone name 3 times it and go on naming very nearly that across the whole range. At 80 dB the notes and products together name the fundamental from −2.0 cents to +2.0 cents and no template at all outside that; the products alone name 1 times it and go on naming very nearly that across the whole range. Equal temperament's major third is +13.7 cents, which is outside the window at both dynamics. The notes alone name the fundamental throughout, because tempering moves every partial of the upper note by the same amount and a template can follow it.

Intervals and chords

A tempered third cannot be one sound with its own ghost

The essay before this one fitted one harmonic template to a just third's played partials and the products of those partials together, and it worked at every dynamic. Tempered, it does not work at all — and not by naming a different note. There is no template. The lower note's partials sit exactly on their harmonics, the upper note's are every one fourteen cents out, and the products are twenty-three cents out the other way, a span of thirty-seven cents across a tolerance of thirty. The window is nine cents wide when the third is soft and two when it is loud, so equal temperament's fourteen falls outside it — while the fifth's two cents fall inside. The consonances a temperament leaves nearly alone are exactly the ones that stay single sounds.

Tuned for the E shape, a cent of intonation is worth about five frets of hand movement. Every way through a ground on E major — I, IV, I, V, IV, I, for 16, 8, 8, 4, 4 and 8 beats — that is the cheapest route at some exchange rate between the piece's mean departure from just and the frets the hand moves, choosing each chord's fingering from its five moveable shapes; once with the guitar tuned to make the E shape exactly just and once on equal-tempered frets. Tuned for the E shape: 12.09 cents for 0 frets; 4.30 cents for 2 frets (worth it above 0.3 frets a cent); 2.09 cents for 10 frets (worth it above 3.6 frets a cent); 0.00 cents for 20 frets (worth it above 4.8 frets a cent). On the frets: 6.85 cents for 0 frets; 6.15 cents for 2 frets (worth it above 2.8 frets a cent); 5.95 cents for 4 frets (worth it above 10.2 frets a cent); 5.78 cents for 10 frets (worth it above 34.5 frets a cent); 5.60 cents for 20 frets (worth it above 57.5 frets a cent). Tuned for the shape, the route that keeps every chord in it is the cheapest once a cent is worth 4.8 frets; on the frets it needs 57.5.

Pitch and tuning

Tuned for one shape, a cent is worth five frets

A guitar can be tuned so that one chord shape is exactly just wherever it is moved on the neck, and every other shape is then between twelve and fifteen cents out — worse than on the frets. So a player of such a tuning chooses, chord by chord, between intonation and the distance the hand travels. Searched exactly over every fingering of a I–IV–V ground, the route that keeps every chord in the one shape is the cheapest as soon as a cent of the piece's mean error is worth 4.8 frets of movement. On equal-tempered frets the same route needs a cent to be worth fifty-seven.

In every key with an open chord, the shape to tune for is that chord's. For a I–IV–I–V–IV–I ground in each of seven major keys, how far the hand moves on the route that keeps every chord exactly just, with the guitar tuned for each of the five moveable shapes, in frets. C: C 20, A 24, G 26, E 28, D 28; D: C 24, A 24, G 20, E 28, D 16; E: C 24, A 20, G 28, E 20, D 24; G: C 24, A 28, G 16, E 24, D 26; A: C 28, A 16, G 24, E 24, D 20; F: C 28, A 28, G 28, E 24, D 24; B♭: C 28, A 24, G 24, E 30, D 28. The keys whose tonic is an open chord — C, D, E, G and A — are served best by the shape of that chord in 5 of 5 cases; F and B♭ have no open tonic, and every shape leaves the hand further from the nut.

Pitch and tuning

Tune a guitar for the shape of its key's open chord

A guitar can be tuned for any of five moveable chord shapes, and a player who then keeps every chord in that shape plays exactly in tune while the hand travels. Which shape should the tuning serve? Searched over seven keys and all five shapes, in every key whose tonic is an open chord the answer is that chord's own shape — C for C, D for D, G for G, A for A, and a tie between the E and A shapes in E. The reason is not the key and not the progression. It is that an open chord needs no barre: its hand sits one or two frets above the nut, and that saving is the whole of the difference.

On a guitar tuned for any major shape, no minor chord is within eight and a half cents of just. Each moveable minor shape — the open E minor, A minor and D minor chords — with its mean departure from just when the guitar is tuned to make one of the five major shapes exactly just, and on equal-tempered frets. E minor shape: 10.30 tuned for C, 10.95 tuned for A, 10.04 tuned for G, 9.78 tuned for E, 10.92 tuned for D; 5.74 on the frets. A minor shape: 10.17 tuned for C, 11.73 tuned for A, 14.47 tuned for G, 8.60 tuned for E, 12.20 tuned for D; 6.65 on the frets. D minor shape: 16.94 tuned for C, 9.78 tuned for A, 10.75 tuned for G, 11.41 tuned for E, 14.66 tuned for D; 8.15 on the frets. The least-bad is the A minor shape on a guitar tuned for the E shape, at 8.60 cents, and every one is worse than the frets.

Pitch and tuning

A tuned guitar has no minor chord in tune

A guitar tuned to make one major chord shape exactly just can play every major chord in tune by moving that shape. It cannot play a minor chord in tune anywhere: every minor shape disagrees with every major shape about some pair of strings by eleven cents or more, and on any of the five tunings the least-bad minor chord — the A minor shape on a guitar tuned for the E shape — is 8.60 cents out, worse than on the frets. Put one minor chord into a I–vi–IV–V ground and the most-in-tune tuning in every key becomes the E shape's, because it is the one that hosts that chord least badly.

With its silent string placed for the minor chord, the C shape's tuning is the most in tune in every key. The I–vi–IV–V ground twice round in five major keys: the mean departure from just of the most-in-tune route under each major shape's tuning, with the strings that shape does not sound left where the one-shape tuning put them and placed for the minor chord. C: C 2.54 → 1.57; A 2.44 → 2.35; G 2.51 → 2.51; E 2.15 → 2.15; D 2.73 → 1.73. D: C 2.54 → 1.57; A 2.44 → 2.35; G 2.51 → 2.51; E 2.15 → 2.15; D 2.73 → 1.73. E: C 2.54 → 1.57; A 2.44 → 2.35; G 2.51 → 2.51; E 2.15 → 2.15; D 2.73 → 1.73. G: C 2.54 → 1.57; A 2.44 → 2.35; G 2.51 → 2.51; E 2.15 → 2.15; D 2.73 → 1.73. A: C 2.54 → 1.57; A 2.44 → 2.35; G 2.51 → 2.51; E 2.15 → 2.15; D 2.73 → 1.73. Under the C shape's tuning every major chord is exact and the minor chord is an E minor shape 6.26 cents out, for a quarter of the piece.

Pitch and tuning

A silent string tunes the minor chord

A continuous search over a guitar's six strings, run on a piece with one minor chord in it, never settles between two chord shapes: at any share of the piece, the best tuning is exact for one chord and makes the other pay. But the tunings the earlier essays called exact for a shape were exact only on the strings that shape sounds. The C shape never sounds its low E string, and tuned to the C chord's own E it hosts an E minor chord at 6.26 cents — not the 8.60 that was said to be the best any tuned guitar could do. The pop ground's most in-tune tuning is then the C shape's, in every key, at 1.57 cents.

Tuned for the whole route, the I–vi–ii–V ground wants no shape at all until the hand is allowed to move. The best tuning for the I–vi–ii–V ground twice round, found by routing and retuning in turn from seventeen starts, in five keys and at eight exchange rates between a cent of mean error and a fret of hand movement: what chord shape it makes exact, its cents from just and the frets its route moves. C major: 1, nothing (6.62 cents, 0 frets); 2, nothing (6.62 cents, 0 frets); 3, nothing (6.62 cents, 0 frets); 5, nothing (6.62 cents, 0 frets); 8, D (4.17 cents, 15 frets); 12, D (4.17 cents, 15 frets); 20, C (3.13 cents, 30 frets); 50, C (3.13 cents, 30 frets). D major: 1, nothing (6.52 cents, 0 frets); 2, nothing (6.52 cents, 0 frets); 3, nothing (6.52 cents, 0 frets); 5, nothing (6.52 cents, 0 frets); 8, D (4.17 cents, 15 frets); 12, D (4.17 cents, 15 frets); 20, C (3.13 cents, 33 frets); 50, C (3.13 cents, 33 frets). E major: 1, nothing (6.62 cents, 0 frets); 2, nothing (6.62 cents, 0 frets); 3, nothing (6.62 cents, 0 frets); 5, nothing (6.62 cents, 0 frets); 8, E (4.60 cents, 15 frets); 12, A minor (4.11 cents, 20 frets); 20, A minor (3.78 cents, 26 frets); 50, C (3.13 cents, 49 frets). G major: 1, nothing (7.71 cents, 0 frets); 2, nothing (7.71 cents, 0 frets); 3, nothing (7.71 cents, 0 frets); 5, E minor (4.71 cents, 9 frets); 8, E minor (4.27 cents, 12 frets); 12, D (3.45 cents, 21 frets); 20, C (3.13 cents, 27 frets); 50, C (3.13 cents, 27 frets). A major: 1, nothing (6.52 cents, 0 frets); 2, nothing (6.52 cents, 0 frets); 3, nothing (6.52 cents, 0 frets); 5, nothing (5.90 cents, 3 frets); 8, C (4.11 cents, 15 frets); 12, C (4.11 cents, 15 frets); 20, C (4.11 cents, 15 frets); 50, C (3.13 cents, 35 frets). At the bottom of the range the best tuning is exact for no chord and the hand does not move; at the top it is one major shape with its silent string placed.

Pitch and tuning

A guitar tuned for a still hand is tuned for nothing

Tune a guitar for a whole route through a ground rather than for one chord shape — routing, retuning and routing again until neither changes — and the answer depends on how much the player is willing to move. Valuing intonation highly, the best tuning is one major shape with its silent string placed, as before. In the middle it can be exact for a minor shape instead, because the route that keeps the hand near the nut plays its minor chords open. And for a player who will not move at all, the best tuning is exact for no chord whatever: every string within two cents of its fret, beating equal temperament by a few tenths of a cent. It is the compromise the earlier search went looking for, and it lives where nobody looked.

The set's tuning is the D shape's on the strings it sounds, and the set's own on the two it does not. The six strings' offsets from equal temperament, low E pinned, of the single tuning found for the I–vi–IV–V ground in five keys at two, five and fifty frets a cent, beside the D shape's placed tuning, dashed. 2: low E +0.0, A +2.0, D +2.0, G +3.9, B +2.0, high E −11.7, exact for D; 5: low E +0.0, A +2.0, D +2.0, G +4.0, B +2.0, high E −11.7, exact for D; 50: low E +0.0, A +13.7, D +0.0, G +15.6, B +13.7, high E +0.0, exact for C. The D shape's placed tuning: low E +0.0, A +2.0, D +8.5, G +10.5, B +8.5, high E −5.2. On the D, G, B and high E strings, measured from the D string, the five-frets tuning and the placed D tuning agree; they differ only on the two strings the D shape leaves silent.

Pitch and tuning

A guitar tuned for a set is tuned for the D shape

Tune a guitar once for an evening of songs in five keys rather than once per song, and the tuning the set wants is not the compromise the single-song search predicted. At the valuation a typical player puts on a cent, it is exact for the D shape — the one fingering every key's route uses for some chord — with that shape's two silent strings placed for the set rather than for any song. Not retuning between songs costs a fifth of a cent. Only a set that mixes two grounds, played by a player who will not move, wants a tuning exact for nothing.

However many keys the set spans, its tuning is exact for a shape — the D shape from four keys on. The I–vi–IV–V ground in a set of songs whose keys are added one at a time round the circle of fifths from C — C, G, D, A, E, B, F♯, D♭, A♭, E♭, B♭, F — at five frets a cent: the set's mean cost with each song given its own tuning, with one tuning for the set, and on equal-tempered frets, with the shape the set's tuning is exact for above each point. 1 key: own 5.17, set 5.17 (C), frets 7.60; 2 keys: own 5.27, set 5.27 (C), frets 7.67; 3 keys: own 5.27, set 5.43 (C), frets 7.54; 4 keys: own 5.01, set 5.33 (D), frets 7.39; 5 keys: own 5.14, set 5.36 (D), frets 7.43; 6 keys: own 5.23, set 5.41 (D), frets 7.46; 7 keys: own 5.29, set 5.47 (D), frets 7.48; 8 keys: own 5.34, set 5.49 (D), frets 7.49; 9 keys: own 5.38, set 5.51 (D), frets 7.51; 10 keys: own 5.41, set 5.54 (D), frets 7.52; 11 keys: own 5.43, set 5.55 (D), frets 7.53; 12 keys: own 5.45, set 5.57 (D), frets 7.51. The set never stops sharing a shape: up to three keys it is the C shape's, and from four to all twelve the D shape's.

Pitch and tuning

One guitar tuning serves every key, and it is the D shape's

A guitar tuned once for a set of pop songs in five keys is tuned exact for the D shape. The open question was the converse: which songs a given tuning serves, and how many keys a set can span before its best tuning stops being any shape's. Grown one key at a time round the circle of fifths, the set never stops sharing a shape: up to three keys it is the C shape's tuning, and from four keys to all twelve it is the D shape's, with the same offsets to a fifth of a cent. The five-key tuning serves the seven keys outside the set better than the two open keys inside it, and not retuning between songs costs less the more keys the evening spans.

Only the D a just ii wants gives the chord a bass that survives its voicing. The bass a pitch mechanism fitting the whole chord — every note's partials and every audible difference and cubic tone — names under the ii chord of C major, with its D tuned three ways, in root position and both inversions, at 50 and 80 dB. the scale's D, 9 : 8: D F A at 50 dB G ±0, D F A at 80 dB none, F A D at 50 dB B +24, F A D at 80 dB F −2, A D F at 50 dB A −30, A D F at 80 dB none. a just ii's D, 10 : 9: D F A at 50 dB B♭ −4, D F A at 80 dB B♭ −4, F A D at 50 dB B♭ −4, F A D at 80 dB B♭ −4, A D F at 50 dB F +13, A D F at 80 dB B♭ −4. equal temperament: D F A at 50 dB G +7, D F A at 80 dB D −1, F A D at 50 dB G +7, F A D at 80 dB D −1, A D F at 50 dB A −23, A D F at 80 dB D ±0. B♭ is the bass the 10 : 9 D implies, since D–F–A is then 10 : 12 : 15 over it.

Intervals and chords

The scale's ii chord borrows the dominant's bass

Heard quietly, the pair D–F of a just C major scale brings the bass of the other D with it — B♭, the bass the 10 : 9 D implies — so the comma is resolved by the listener rather than by the players. Add the A that makes the pair a ii chord and that stops being true. The whole-object fit now hears the chord over G, as the top of a dominant ninth, and it does so because the template prefers simple harmonics to accurate ones; a different bass is heard in each inversion, and at a forte none at all. Only a ii built on the 10 : 9 D has one bass in every voicing and at every dynamic. Played quietly from a fixed scale, I–vi–ii–V–I is heard with the bass line C, F, G, G, C: the ii merged into the chord after it, and the progression home.

A set in twelve keys keeps its shape through any number of rock songs and loses it to two minor ones. The I–vi–IV–V ground in all twelve keys, with songs outside it added one key at a time — minor-key songs on A, E, B, F♯ and on round the circle, or rock songs on G, D, A, E and on — at five frets a cent: the set's mean cost under its one best tuning, with the shape that tuning is exact for above each point. minor-key songs, i–iv–V–i: 0 added 5.57 (D), 1 added 5.77 (D), 2 added 5.97 (C), 3 added 6.01 (C), 4 added 6.04 (C), 6 added 6.08 (C), 8 added 6.12 (C), 12 added 6.03 (E minor). rock songs, I–♭VII–IV–I: 0 added 5.57 (D), 1 added 5.49 (D), 2 added 5.42 (D), 3 added 5.36 (D), 4 added 5.22 (D), 6 added 5.14 (D), 8 added 5.09 (D), 12 added 4.96 (D).

Pitch and tuning

Two minor songs retune a guitar set for their dominant

A guitar tuned once for an evening of pop songs in every key is tuned exact for the D shape, because every key's route through the ground passes through it. Add songs that are not that ground and the tuning holds or breaks by a rule nobody would guess. Rock songs on I, ♭VII and IV never move it off the D shape, however many are added: they retune only the two strings the D shape leaves silent. One minor-key song leaves it alone too. Two move it to the C shape's tuning, and the chord they move it for is not one of their minor chords but their one major chord — the dominant, fingered in the C shape. Twelve move it again, to the E-minor shape's.

What each minor-key song gains from the C shape's tuning. For a minor-key song, i–iv–V–i, in each of the twelve keys, the saving in cents plus frets over five from playing it under the C shape's tuning rather than the twelve-key ground's own, the D shape's: C minor 1.69, C♯ minor 1.69, D minor 1.69, E♭ minor 1.69, E minor 1.69, F minor 1.69, F♯ minor 1.69, G minor 0.33, A♭ minor 1.69, A minor 1.69, B♭ minor 1.69, B minor 1.61. Switching the tuning costs the twelve ground songs 3.90 between them, so the switch pays when the added songs save more than that: the lines mark a half and a third of it, what each of two or three songs would have to save. Ten keys save 1.69 each — less than half, more than a third. G minor saves 0.33, because under the ground's tuning its dominant is the open D chord, already exact.

Pitch and tuning

A guitar set keeps its tuning until the third minor song

Two minor-key songs were found to move a pop set's guitar tuning from the D shape to the C shape. Priced as a vote — what the move costs the twelve ground songs, against what it saves each minor song — two songs do not pay for it and three do. The earlier search had never tried keeping the tuning the set already had. And the key a minor song is in matters only at the nut: ten keys vote alike, and G minor, whose dominant is the open D chord, hardly votes at all.

The minor songs' tonic takes the tuning only at the ninth minor song. The twelve-key ground with minor-key songs added on A, E, B, F♯ and on round the circle: the set's mean cost, in cents plus frets over five, under three fixed tunings — the ground's own (the D shape's), the C shape's (the minor songs' dominant exact) and the E-minor shape's (their tonic exact). 0: 5.567, 5.892, 6.499; 1: 5.767, 5.937, 6.436; 2: 5.938, 5.975, 6.327; 3: 6.081, 6.008, 6.285; 4: 6.211, 6.037, 6.247; 5: 6.326, 6.062, 6.214; 6: 6.428, 6.085, 6.185; 7: 6.520, 6.105, 6.159; 8: 6.602, 6.123, 6.135; 9: 6.638, 6.102, 6.076; 10: 6.707, 6.118, 6.058; 11: 6.711, 6.134, 6.042; 12: 6.772, 6.148, 6.028. The ground's tuning is cheapest to two songs, the C shape's from three to eight, and the E-minor shape's from nine — when minor songs are nine of twenty-one.

Pitch and tuning

A guitar set tunes for its minor chords only when nearly half its songs are minor

Add a third candidate to the vote — the tuning that makes the minor songs' own tonic chord exact — and a pop set's guitar passes through three tunings as minor songs are added: the ground's to the second, the dominant's from the third, and the tonic's only from the tenth, when ten songs in twenty-two are minor. Nothing in between is ever chosen: the two minor-song tunings differ on two strings, and every compromise between them costs the ground more than either end. Two keys break the pattern at the nut, and one earlier figure turns out to be wrong by a song.

Intervals and chords

A stiff string slows the fourths and quickens the fifths

Every rate in a bearing plan is computed from whole-number partials, and a piano string's partials are not whole numbers. The correction is usually summarised as 'every real beat is faster than the table'. It is not: stiffness quickens the beat of every narrow interval and slows the beat of every wide one, and on a small upright the tempered fourth stops beating altogether. Both errors push the same way, so a tuner who counts the printed table leaves every interval wide, and a fifth and a fourth counted that way make an octave stretched to the 4:2 type or a little past it.

Pitch and tuning

One retuning is worth what the pop songs pay for the minor tuning

A guitarist allowed to retune once in an evening plays the pop songs in their own tuning and the minor songs in theirs. What that saves rises with every minor song until the tenth, and then stops: from there on it is exactly what the twelve pop songs would have paid to be played in the minor songs' tuning, and adding minor songs cannot raise it. The C shape's tuning, which a one-tuning guitar spends most of the range in, is never used. And the whole saving belongs to the running order. Put the same ten minor songs in two runs instead of one and a single retuning is worth three-fifths as much; spread them through the evening and it is worth a fifth.

The fourth below a note and the fifth above it differ by the same amount wherever the note is put. The fourth A3–D4 and the fifth D4–A4 on the grand modelled here, each note with its own string's stiffness, with the octave 1.9 cents wider than 2:1 in its fundamentals: the beat rate of each, signed so that a wide fourth and a narrow fifth are both positive, as the middle note is moved three cents either side of its tempered place. Both lines move together, and the fourth always beats 0.15 a second faster than the fifth — the difference between the upper note's second partial and the lower note's fourth, which the middle note does not touch. The two beat alike only when the octave is 1.60 cents wide, the octave whose 4:2 member is silent.

Intervals and chords

The fourth–fifth check cannot hear the note it checks

A tuner tests a temperament by sounding the fourth below a note and the fifth above it, inside one octave, and comparing their beats. Both beats run to the same partial of the middle note, so moving that note speeds or slows both together and leaves their difference untouched. The check is a test of the octave, and of one particular octave: the 4:2. Measured with each note's own stiffness, the printed table laid fourth first builds exactly that octave on any piano; laid fifth first it overshoots, and the check pulls it back by a third of a cent.

Each family of checks builds its own octave, and the thirds build one twice as wide as the fourths and fifths. On each lower note from F3 to F4, on the grand modelled here: how many cents wider than 2:1 the octave lands when it is filled by a chain of intervals each counted to the rate a bearing plan prints — a fourth then a fifth, three major thirds, a major sixth then a minor third, four minor thirds — against the 4:2, 6:3 and 8:4 octave types, each note with its own stiffness. At A3: fourth, then fifth 1.60, three major thirds 3.20, minor third, then major sixth 3.57, four minor thirds 8.09; types 1.60, 3.56 and 6.22. The fourth and fifth build the 4:2 octave exactly and the minor third and major sixth the 6:3 within a few hundredths of a cent; three major thirds build an octave 2.00 times as wide as the fourth and fifth's, just short of the 6:3.

Intervals and chords

The thirds and the fourths check different octaves

Every interval counted to its printed beat rate on a stiff string is left wide, by an amount set by the squares of the partial numbers that meet in it. Add the links of a chain and the sum lands on an octave type: a fourth and a fifth make the 4:2, a minor third and a major sixth make the 6:3, and three major thirds come just short of the 6:3. So the two most familiar checks in a bearing are exact tests of two different octaves, and on a stiff piano no octave passes both. Each fails on the other's octave by a beat a second at A3.

Pitch and tuning

The minor half's own tuning belongs to one key

A guitarist who retunes once between the pop songs and the minor songs has been choosing among three tunings found for whole evenings. Searched for the minor songs alone, the best tuning is none of them — but only in A, B and D minor, where it gives the tonic and subdominant one shape at the nut and both are in tune. In A it saves six-tenths a song, and the refund a retuning buys stops having a ceiling. In the other nine keys, and in any minor half that mixes keys, the E-minor shape's tuning was already the answer.

With a capo up to fret 7, 8 of the twelve minor keys are played as B minor in one tuning. A minor-key song, i–iv–V–i, in each of the twelve keys, in cents plus frets over five, under the tuning searched for the song on A (the B string raised 13.7 cents): at the nut, and at the capo position up to fret 7 that suits it best, with the E-minor shape's tuning at its best capo for comparison. C minor 5.72 → 4.53 (capo 1); G minor 5.72 → 5.72; D minor 5.72 → 4.53 (capo 3); A minor 5.09 → 5.09; E minor 5.72 → 4.53 (capo 5); B minor 4.53 → 4.53; F♯ minor 5.72 → 4.53 (capo 7); C♯ minor 5.72 → 4.53 (capo 2); A♭ minor 5.72 → 5.72; E♭ minor 5.72 → 4.53 (capo 4); B♭ minor 5.72 → 5.09 (capo 1); F minor 5.72 → 4.53 (capo 6). 8 keys reach 4.53, B minor's price, by putting the capo where B minor's open-string chords become theirs. E minor costs 4.53 at capo 5, against 4.91 in the E-minor shape's tuning.

Pitch and tuning

A capo is a transposition the tuning cannot see

Tempered frets make a chord shape's error the same wherever it sits, so a song with a capo at the fifth fret is, to the tuning, the song five semitones lower on a shorter neck. That lets every minor key borrow B minor's open-string fingerings, and eight of the twelve take the raised-B tuning's best price with a capo no higher than the seventh fret. A minor half in mixed keys, which gave a retuning nothing, now nearly doubles its refund — but the capo pays the single-tuned evening as well, and a minor half all in A gets less from its retuning than before.

With the capo left at fret 5, E minor becomes B minor and the other keys pay for it. Twelve minor-key songs, i–iv–V–i, one in each key round the circle, in the raised-B tuning, in cents plus frets over five: at the nut, with the capo left at fret 5 for every song, and at each song's own best capo up to fret 7. C minor 5.72, 8.62 at fret 5, 4.53 at its own; G minor 5.72, 5.72 at fret 5, 5.72 at its own; D minor 5.72, 5.09 at fret 5, 4.53 at its own; A minor 5.09, 6.13 at fret 5, 5.09 at its own; E minor 5.72, 4.53 at fret 5, 4.53 at its own; B minor 4.53, 8.62 at fret 5, 4.53 at its own; F♯ minor 5.72, 5.72 at fret 5, 4.53 at its own; C♯ minor 5.72, 8.62 at fret 5, 4.53 at its own; A♭ minor 5.72, 5.72 at fret 5, 5.72 at its own; E♭ minor 5.72, 5.72 at fret 5, 4.53 at its own; B♭ minor 5.72, 7.82 at fret 5, 5.09 at its own; F minor 5.72, 5.72 at fret 5, 4.53 at its own. The twelve cost 66.82 at the nut, 78.04 with the capo at fret 5 and 57.87 with it moved for every song. At fret 5 the song on E takes B minor's fingerings and the song on D A minor's; every other key is moved to a key the tuning does not favour, on a neck 5 frets shorter.

Pitch and tuning

A capo left in place serves one key

Moved for every song, a capo lets nearly every minor key borrow B minor's open-string fingerings. Left at one fret for a whole run of minor songs, as most guitarists leave it, it does something much weaker: it renames the keys. Over a run in every key that is a reshuffle on a shorter neck and can never beat the nut; over the four keys guitarists use most it gains nothing either, because two of them are already the tuning's best. A fixed capo pays only a run in very few keys, spaced the way the tuning's cheapest keys are.

Pitch and tuning

A set ordered by key relation moves the capo most

Grouped by the fret each song wants, a set of twelve minor songs in E, A, D and B gets the whole of what a moving capo saves with three capo changes. A real set is not grouped by fret; it is ordered for the music, and the commonest rule for that order is key relation — neighbouring songs a fifth apart. That order is the worst the capo can face: no two neighbours want the same fret, and three changes recover thirty-five per cent of the saving against fifty-one for a random order. The capo groups keys a tone apart, and a set built on fifths is the order that keeps them furthest from each other.

Pitch and tuning

A set steps by fifths where the capo moves

A set of twelve minor songs in E, A, D and B can be ordered for the capo or for key relation, and the two orders pull apart. Searched over every one of its 41,304 orders, the trade between them is exact and simple: a set can put a fifth between neighbouring songs at each capo change for nothing, because a fifth always crosses frets and a capo change is where crossing is paid for. Beyond that each extra fifth costs about seven points of the capo's saving. The tone steps the earlier essay suspected of carrying the compromise are where the capo stays still.

A meantone keyboard is a stretch of the chain of fifths, and every split key lengthens it by one place at an end. 14 keys hold the positions from A♭ to D♯. That is 10 major and 10 minor triads; major keys E♭, B♭, F, C, G, D, A, E; minor keys with a raised leading note C, G, D, A, E.

Pitch and tuning

A split key buys one of everything, and two buy C and E

An organ builder with money for two extra keys could have put them anywhere, and the arithmetic of triads cannot say where. On a meantone keyboard every added key buys exactly one major triad, one minor triad, one major key and one minor key, wherever it goes. So the count is no guide, and the choice was about which keys. The two accidentals most often divided, G♯/A♭ and D♯/E♭, are one on each side of the chain, and they are exactly what completes the minor keys on C and E. What they buy by the roughness sum is seven per cent.

All themes